Questions tagged [integration]
For questions about the properties of integrals. Use in conjunction with (indefinite-integral), (definite-integral), (improper-integrals) or another tag(s) that describe the type of integral being considered. This tag often goes along with the (calculus) tag.
70,172
questions
6
votes
1
answer
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How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?
I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion.
And its analytic ...
- 63
2
votes
1
answer
3k
views
Indefinite integral with Euler's number
Trying to solve this tricky one:
$$\int {e^{2x}} \sqrt{1 + e^{2x}}dx$$
I am pretty sure I need to use integration by parts, so I have come up with this so far:
$$u=e^{2x} \Rightarrow du=2e^{2x}$$ $$...
- 1,363
2
votes
1
answer
1k
views
Fourier transform in Mathematica
When I calculate the Fourier transform of the function $$f(t) = \mathrm e^{-|t|/\tau} \text{ with } \tau >0$$ in Mathematica once via the function FourierTransform and once by hand, I get different ...
2
votes
1
answer
204
views
Primitive to a function. Is there one?
Is there a primitive function to:
$$\int \! \frac{\int \! \frac{\ln(x+1)\, \mathrm{d} x}{x}\, \mathrm{d} x}{x}$$
- 81
3
votes
3
answers
427
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$f(x)$ monotonic integrable function and $\lim_{x\to \infty}\frac{1}{x}\int_{0}^{x}f(t)dt=a$, Prove: $\lim_{x\to \infty}f(x)=a$
Let $f(x)$ be a monotonic increasing function on $[0,\infty)$ and for every $x>0$ it is integrable in $[0,x]$, so that $\lim_{x\to \infty}\frac{1}{x}\int_{0}^{x}f(t)dt=a$. I need to prove that $\...
- 6,890
6
votes
1
answer
876
views
How do I express $\int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz$ in terms of named functions?
Recently I derived an expression for a particular probability density function. The expression contains the integral
$$
f(t,v,a) = \int_0^t \frac{{\rm e}^{-a^2 z}}{\sqrt{z} (z+v)} \,dz =
2a \int_0^{...
- 508
42
votes
4
answers
2k
views
Is the integral $\int_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$ equal for all $a \neq 0$?
Let $a$ be a non-zero real number. Is it true that the value of $$\int\limits_0^\infty \frac{\mathrm{d} x}{(1+x^2)(1+x^a)}$$ is independent on $a$?
- 1,523
2
votes
1
answer
945
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Continuity and $L^p$ spaces
I have been wondering how to solve this question I saw in a textbook. Given $ g \in \bigcup _{1\leq p\leq \infty} L^{p}$ define, for $ r \in [ 0,1]$ , $$ G(r) = \int_{0}^{r} g(t) dt \;.$$ Show that $G$...
- 23
0
votes
1
answer
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Contour integration help: $\int_{\gamma}e^zdz$ with $\gamma(\theta)$=$e^{i\theta}$
I am stuck integrating $$\int_{\gamma}e^zdz$$
with $\gamma(\theta)$=$e^{i\theta}$ and $0\le\theta\le2\pi$.
I got up to $$\int_{\gamma}e^zdz = \int_{0}^{2\pi}e^{e^{i\theta}}\cdot ie^{i\theta}d\theta = \...
- 97
1
vote
1
answer
149
views
Trouble with basic integration
I'm doing the something that results in following integral:
$$f(z) = \int_{-\infty}^\infty \frac{1}{2\pi}x\exp\left(\frac{-x^2}{2}\left(1+z^2\right)\right) dx$$
Then since $f(z)$ is even we get:
$$f(z)...
- 822
1
vote
5
answers
230
views
Help with integrating $\int \frac{t^3}{1+t^2} ~dt$
What am I doing wrong on this integration problem?
$$
\begin{align*}
\int\frac{t^3}{1+t^2} &=
\frac14 t^4 (\ln(1+t^2) (t+\frac13 t^3))
\\ &= \frac14 t^4(t \ln(1+t^2)+\frac13 t^3 \ln(1+t^...
- 1,073
5
votes
1
answer
83
views
Continuous functions question
I am stuck on the problem:
Find all continuous functions $h$ satisfying
$$\int_{0}^{x}h(y)dy=\left [ h(x) \right ]^{2}+C$$
for some constant $C$.
- 51
0
votes
1
answer
89
views
Proving that an integral means "potential" in physics terms
It's possible to prove with mathematical terms using a part of the physics knowledge this general assumption? I have several formulas in mind but not a general and big picture for this property of the ...
- 271
1
vote
4
answers
173
views
Computing the derivative of $H(z)= \int_{e^z}^{\cos z} \ln(w^3) ~dw$
I am stuck on this problem:
Find the derivative of the function: $$H(z)= \int_{e^z}^{\cos z} \ln(w^3) ~dw.$$
- 343
1
vote
1
answer
207
views
Complicated "functional integral"
I came across the following "functional" at work:
$$
\Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda
$$
it's part of an optimization problem that tries to find $b$,...
- 1,787
2
votes
3
answers
172k
views
How to find the area of the region, bounded by various curves?
Find the area of the region bounded by the curves $y=x^2$ and $y=x$.
Find the area of the region bounded by the curves $y=x^2+1$ and $y=2$
I have a ton of questions like this and I have been graphing ...
- 1,495
0
votes
1
answer
85
views
How to prove this zeta function?
Prove that $\sum_{n=2}^{\infty} \frac{z^{n-1}}{\alpha(n-1)+1}$ is equivalent to $\frac{1}{\alpha} \displaystyle \int_{0}^{1}{ \frac{z t^{\frac{1}{\alpha}}}{1-tz}} dt$?
- 214
0
votes
1
answer
158
views
Calculate the integral $ \int_{0}^{1}(ax^2+bx+c)^{-3/{2}}dx $
How to calculate the integral below:
$$
\int_{0}^{1}(ax^2+bx+c)^{-3/{2}}dx
$$
user20273
0
votes
3
answers
2k
views
find the area of the largest rectangle that can fit inside a semi circle of radius 2 cm
find the area of the largest rectangle that can fit inside a semi circle of radius 2 cm
I have absolutely no idea where to get started on this...What I did do is
$A=(\pi r^2)/2$ (its a semi circle)
...
- 1,495
1
vote
2
answers
1k
views
calculus integration question
We have a weekly assignment and the teacher posts solution but doesn't EXPLAIN how she got the answer. just gives you the answer.
So I got this question wrong and I need help on how the answer was ...
- 1,495
14
votes
2
answers
1k
views
Regularization and $\int_{0}^{\infty}\sin x \;\mathrm{d}x$
In my grad quantum/E&M classes I had to do intuition-bending regularization of integrals that didn't seem mathematically justified (but got full credit and were repeated in the solutions) like the ...
- 993
1
vote
1
answer
479
views
Contour Integral with confluent hypergeometric function
Can we get a closed form for the following contour integral?. Let us assume that n is a non-negative integer,
$\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}\frac{\Gamma(n-s)\Gamma(s)\Gamma(k-s)}{\...
- 543
2
votes
2
answers
110
views
Help with finding integral
I've been trying for an embarrassingly long time to figure this one out. It looks like it should crack under integration by parts and integration by substitution, but I am having trouble with it. ...
- 634
3
votes
1
answer
978
views
Cutting off divergent integrals
In Quantum Field Theory, one has to deal with loop integrals which are divergent. The way out is to regularise your integral. For example one can introduce a cutoff, like in the Pauli-Villlars ...
- 131
6
votes
2
answers
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views
If $f$ and $g$ are integrable then is $\max\{f,g\}$? [duplicate]
Possible Duplicate:
Is the pointwise maximum of two Riemann integrable functions Riemann integrable?
Let $f$ and $g$ be two integrable real functions. Is this leads that $\max\{f,g\}$ is integrable ...
- 1,523
15
votes
4
answers
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views
How do you integrate imaginary numbers?
How would you find, for instance, $\int_0^4 i\> x \,dx$? Can you just treat $i$ as a constant, or do you have to do something more sophisticated?
Thanks!
- 3,965
2
votes
1
answer
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If $f(x)$ and $g(x)$ are Riemann integrable and $f(x)\leq h(x)\leq g(x)$, must $h(x)$ be Riemann integrable?
Let $f$ and $g$ be Riemann integrable (real) functions and
$$f(x)\leq h(x)\leq g(x).$$
Is it true that $h(x)$ is Riemann integrable? Can someone post a proof (if there is)?
Thanks.
- 1,523
2
votes
1
answer
444
views
About integration related to the gamma function
I would like to compute the integral
$$
\int_{0}^{\infty}\frac{1}{\sqrt{2t}}e^{-\frac{1}{2t}}dt
$$
which wolfram alpha says that it does not converge. However by letting $x=1/2t$ I get $dt=\frac{-...
- 23
8
votes
1
answer
2k
views
Relation between integral by parts and Fubini's theorem
In probability, I have seen some examples for which both Fubini's theorem and integration by parts (for Riemann-Stieltjes integrals with cdf as integrator) provide different but correct solutions. For ...
- 45.4k
1
vote
2
answers
453
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If $f$ is integrable in $[a-1,b+1]$, then prove that $\lim \limits_{h\to 0} \ \int_{a}^{b}|f(x+h)-f(x)| ~ dx=0$
I'd really appreciate your help with the following problem:
Let $f$ be a integrable function in $[a-1,b+1]$. I need to prove that:
$$\lim_{h\to 0} \int_{a}^{b} |f(x+h)-f(x)|dx=0 .$$
Basically ...
- 6,890
3
votes
1
answer
180
views
How to solve this: $\int \frac{1}{(5-x-x^{2})^{5/2}}dx$ using a trigonometric substitution
I've completed the squares in order to get a fraction in the integrand of the form $\frac {1}{\sqrt{a^2-x^2}}$ that can be easily substituted by a trigonometric function (drawing the respective ...
- 287
3
votes
2
answers
1k
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Integration in Various Metric Spaces
How does the integral of f(x) with respect to x in $R^2$ change if a non-Euclidean metric is used? For instance, let f(x)=$x^2$. Would the value of a definite integral along an interval change?
- 2,620
-1
votes
2
answers
130
views
Are there other ways to calculate integrals or a way to keep all these formulas apart?
I'm having quite a hard time keeping apart all the different formulas I need to calculate integrals. This is why I'm wondering if there is another way to calculate them, or even a way to keep all ...
- 627
2
votes
1
answer
603
views
How to solve $ \displaystyle\int \bigg(\small\sqrt{\normalsize x+\small\sqrt{\normalsize x+\sqrt{ x+\sqrt{x}}}}\;\normalsize\bigg)\;dx$?
How to solve this:
$$\displaystyle\int \bigg(\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\small\sqrt{\normalsize x +\sqrt{x}}}}\;\normalsize\bigg) \;dx$$
- 39
16
votes
1
answer
936
views
challenging integral involving $\zeta(5)$
I ran across a curious integral that seems to be rather tough that some on the site may enjoy.
Show that $$\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^{2}}}{1-x^{2}\sin^{2}(x)}dx = \frac{5\sqrt[5]{{\...
- 13.7k
2
votes
1
answer
541
views
Exponential integration rule
Given the function $e^ae^{-ba}$
The indefinite integral is $\int e^ae^{-ba} \mathrm da = \int e^{a-ba} \mathrm da$ is $\frac{e^ae^{-ba}}{1-b}$
I get that $\int e^u = e^u\frac{\mathrm du}{\mathrm da}$...
- 285
4
votes
3
answers
2k
views
Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?
Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$
Proof (attempt):
We know $-|f|\le f \le|f|$,
so $\int-|f| \le \int f \le \int|f|$.*
Since, if $-b<a<b$, we say $|...
- 2,106
3
votes
1
answer
776
views
Help with this integration by parts boundary term
I have this in my notes:
$\int_\Omega{(\partial_i^2 u)(\partial_k^2 u)}
= \int_{\partial\Omega}{(\partial_i u)(\partial_k^2 u)\cdot n \mathbf{d}S} - \int_\Omega{(\partial_i u)(\partial_i \partial_k^2 ...
- 838
0
votes
3
answers
138
views
Finding the derivative of two integrals and establish their equality.
For $a > 0$, I need to find the derivatives of $F(x) = \int\nolimits_{1}^{x}\frac{dt}{t}$ and of $G(x) = \int_{a}^{ax}\frac{dt}{t}$, $x\in \left ( 0,\infty \right )$ and use them to prove that $G(...
- 119
2
votes
4
answers
3k
views
Prove integral inequality
Assume that a function $f$ is integrable on $[0, x]$ for every $x > 0$.
Prove that for any $x > 0$, $\displaystyle\left (\int_{0}^{x}fdx \right )^2\leq x\int_{0}^{x}f^2dx$.
I have no idea ...
- 119
1
vote
1
answer
166
views
does this integration converge?
Consider the first kind Bessel function $J_0$, one way to define it is $J_0(x)=1/\pi \int_0^\pi \cos(x \sin t)\;dt$.
My question is, $\int_0^n J_0(x)\;dx$ converge when $n$ tends to infinity?
For ...
- 216
1
vote
0
answers
116
views
Continuity of the Upper and Lower Integral
I'd appreciate it if someone could lead me through the steps for this problem:
Let $f$ be defined and bounded on $[a,b]$. Define a function $g$ on $[a,b]$ by the formula $\overline{I}(\chi_{[a,x]} f)$...
- 23
2
votes
1
answer
934
views
Proof of the following fact: $f$ is integrable, $U(f,\mathcal{P})-L(f,\mathcal{P})<\varepsilon$ for any $\varepsilon>0$
From Spivak's Calculus,
For the theorem:
If $f$ is bounded on $[a,b]$, then $f$ is integrable on $[a,b]$ if and only if
for every $\varepsilon > 0$ there is a partition $\mathcal{P}$ of $[a,b]...
- 2,106
2
votes
2
answers
424
views
Integral over the full support of a square and cube of a convolution of normal and uniform
I've got a uniform random variable $X\sim\mathcal{U}(-a,a)$ and a normal random variable $Y\sim\mathcal{N}(0,\sigma^2)$. I am interested in their sum $Z=X+Y$. Using the convolution integral, one can ...
- 5,276
1
vote
2
answers
498
views
Numerical solution of the Dirichlet problem for unit circle in two dimensions
How to approach the problem using integral equations and solve it with numerical methods in matlab?
Are there any tutorials with code for begginers?
- 121
2
votes
1
answer
1k
views
Transformation of Random Variables and Expectation
Suppose we have $0\le\,X\le\,\infty$, and we have $Y = a+b\, X\,\quad$ where $a\in\mathbb{R}$, $b\in\mathbb{R}$, $X \sim \chi^2_\nu(\beta)\quad$ with a PDF $f_X(x)$. Now, I found that the PDF of Y is ...
- 543
14
votes
2
answers
480
views
tough integral involving the Cosine integral
I ran across an integral on a German math site that has a friend of mine and I quite stuck.
They give, without derivation,
$$\int_0^\infty \mathrm{Ci}(\alpha x)\mathrm{Ci}(\beta x)dx=\frac{\pi}{2 \...
- 13.7k
1
vote
1
answer
6k
views
Finding total mass of a plate using Integration
I am having difficulty parsing this question and visualizing how the picture looks and what tools to use to solve
Question: A flat metal plate is in the shape determined by area by the area under the ...
- 1,363
4
votes
1
answer
3k
views
What is the volume of this ring-like solid?
We're learning about triple integrals and such in class.
Here's one of the problems I'm working on:
A cylindrical drill with radius 3 is used to bore a hole through the center of a sphere of radius ...
user13327
0
votes
1
answer
92
views
Area between curves and change of sign
I am wondering ifmy curves look like
$y=9-x^2, z=x^2-3x$
for area between curves, why isit just
$\int^{3}_{-3/2}(9-x^2)-(x^2-3x) dx$
I don’t care if there’s a change of sign?
- 4,383